Designing scattering-free isotropic index profiles using phase-amplitude equations

Abstract

The Helmholtz equation can be written as coupled equations for the amplitude and phase. By considering spatial phase distributions corresponding to reflectionless wave propagation in the plane and solving for the amplitude in terms of this phase, we designed two-dimensional graded-index media which do not scatter light. We give two illustrative examples, the first of which is a periodic grating for which diffraction is completely suppressed at a single frequency at normal incidence to the periodicity. The second example is a medium which behaves as a "beam shifter" at a single frequency; acting to laterally shift a plane wave, or sufficiently wide beam, without reflection.

Case study: A simple optical inverse problem from a geometrical optics point of view

“…We expect that this work contributes to further progress of scattering-free medium design to manipulate light arbitrarily. 6,7 More aspects about this inverse problem of GO are explorable from the perspective of designing a curved refraction surface.…”

“…Moreover when the impedancematching condition is relaxed, the space of possible inhomogeneous media becomes larger to explore by making use of Maxwell's equations. [4][5][6][7] It is curious to note that the analytic tool set of transformation optics and its derivatives build themselves largely on the basis of geometrical optics (GO), which approximates wave optics and reduces the field distribution for electromagnetic waves into the eikonal functions along the trajectories for light rays. 8,9 This demonstrates the power of working on the GO counterpart to simplify the optical inverse problem, which itself is more complicated than the former.…”

Light Confinement by Local Index Tailoring in Inhomogeneous Dielectrics

“…The equation can also be reformulated as an equation for at a given (see ref. [24]) however we do not follow this approach here. An equivalent formulation of the problem, which does not require the numerical solution of Equation (), is given in Section 3 of the Supporting Information.…”

“…Equation () stems from the fact that the designed medium is Hermitian, and a related equation in ref. [24] was aptly named the “energy conservation condition”. The coefficients are calculated from the numerical solution of the Helmholtz equation for the reference medium.…”

“…By establishing a connection between fields in an inhomogeneous reference medium and a design medium with equal scattering coefficients, our optical design technique generalizes the interesting results of refs. [23,24] where the amplitude manipulation was limited to homogeneous reference media only, which considerably simplified the problem.…”